Vector Differentiation

In Engineering Mathematics, vectors are used to represent physical quantities that have magnitude and direction, such as displacement, velocity. Often, these vectors change with time or other variables.

Vector differentiation is the process of finding the derivative of a vector function with respect to a scalar variable, usually time.

If A, B, and C are differential vector functions of scalar u and Φ is a differential scalar function of u, then:

• d/du (A + B ) = dA/du + dB/du
• d/du (A . B ) = A.dB/du + dA/du.B
• d/du (A x B ) = A x dB/du + dA/du x B
• d/du(𝛷𝐀) = 𝛷d𝐀/du + d𝛷/du A
• d/du (A . B x C ) = A .B x dC/du + A . dB/du x C + dA/du. B x C
• d/du(A x ( B x C)) = A x (B x dC/du + A . dB/du x C + dA/du x (B x C)'

Dot product

The dot product gives us a scalar that measures how much one vector extends in the direction of another.

If \vec{A} and \vec{B} are two vectors, then:

\vec{A} \cdot \vec{B} = |\vec{A}| \, |\vec{B}| \, \cos \theta

Where,

• θ- the angle between them

Cross Product

The cross product gives a vector perpendicular to the plane of two vectors.

If \vec{A} and \vec{B} are two vectors, then:

\vec{A} \times \vec{B} = |\vec{A}| \, |\vec{B}| \, \sin \theta \, \hat{n}

Where,

• \hat{n} - Unit vector normal to the plane.

Gradient

The gradient of a scalar function gives the direction and the rate of its maximum increase.

For a scalar field f(x, y, z),

\nabla f = \frac{\partial f}{\partial x} \, \hat{i} + \frac{\partial f}{\partial y} \, \hat{j} + \frac{\partial f}{\partial z} \, \hat{k}

It converts a scalar function into a vector field, used in heat and potential flow problems.

Divergence

Divergence measures how much a vector field spreads out from a point.

\nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}

It is a scalar quantity used in fluid flow and Gauss's law.

Curl

Curl measures the rotation or swirling strength of a vector field.

\nabla \times \vec{A} =\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\A_x & A_y & A_z\end{vmatrix}

Laplacian

Laplacian gives the rate at which rate at which function's value spreads out from a point.

For scalar f(x,y,z):

\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

Solved Questions on Vectors Identities

Question 1: Find the gradient of f(x, y, z) = x²y + yz³.

Solution:

\nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k}

= (2xy) \hat{i} + (x^2 + z^3) \hat{j} + (3yz^2) \hat{k}

Question 2: Find the Divergence of A = \vec{V} = xy \, \hat{i} + yz \, \hat{j} + zx \, \hat{k}

Solution:

Divergence is defined as:

\nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}

Identify components

A_x = xy, A_y = yz, A_z = zx

Compute partial derivatives

\nabla \cdot \vec{A} = = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}

Wait: A_z = zx so ∂A/z∂z = x

\nabla \cdot \vec{A} = y + z + x = x + y + z

Question 3: Find the Curl of \vec{A} = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}

Solution:

Curl is defined as:

\nabla \times \vec{A} =\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\A_x & A_y & A_z\end{vmatrix}

Expand determinant:

\nabla \times \vec{A} =\hat{i} \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right)- \hat{j} \left( \frac{\partial A_z}{\partial x} - \frac{\partial A_x}{\partial z} \right)+ \hat{k} \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right)

Compute deriatives:

\frac{\partial z^2}{\partial y} = 0, \frac{\partial y^2}{\partial z} = 0

Subtract all componets 0

∇ × \overrightarrow{A} = 0

Question 4: Find the Laplacian of f(x,y,z) = x³ + y³+ z³.

Solution:

Laplacian is defined as:

\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

Compute second derivatives:

\frac{\partial^2 f}{\partial x^2} = 6x, \quad\frac{\partial^2 f}{\partial y^2} = 6y, \quad\frac{\partial^2 f}{\partial z^2} = 6z

Add them

∇²f = 6x + 6y + 6z

Unsolved Questions on Vector Identities

Question 1: Find the gradient of the scalar field: f(x, y, z) = e^xysin⁡z

Question 2: Compute the Laplacian of the scalar field: f(x, y, z) = ln⁡(x² + y² + z²).

Question 3: Compute the divergence of the vector field: \vec{A}(x,y,z) = x^2 y \, \hat{i} + y z^2 \, \hat{j} + x z \, \hat{k}

Question 4: Calculate the curl of the vector field: \vec{A}(x, y, z) = y^2 \, \hat{i} + xz \, \hat{j} + yz \, \hat{k}